let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite AMI-Struct of N
for F being NAT -defined the Instructions of b₁ -valued FinPartState of st F is closed holds
F is really-closed
let S be non empty stored-program IC-Ins-separated definite AMI-Struct of N; :: thesis: for F being NAT -defined the Instructions of S -valued FinPartState of st F is closed holds
F is really-closed
let F be NAT -defined the Instructions of S -valued FinPartState of ; :: thesis: ( F is closed implies F is really-closed )
assume A1: F is closed ; :: thesis: F is really-closed
let s be State of S; :: according to AMISTD_1:def 18 :: thesis: ( IC s in dom F implies for k being Element of NAT holds IC (Comput (F,s,k)) in dom F )
assume A3: IC s in dom F ; :: thesis: for k being Element of NAT holds IC (Comput (F,s,k)) in dom F
defpred S₁[ Element of NAT ] means IC (Comput (F,s,$1)) in dom F;

A4: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A5: S₁[k] ; :: thesis: S₁[k + 1]
reconsider t = Comput (F,s,k) as Element of product the Object-Kind of S by PBOOLE:155;
set l = IC (Comput (F,s,k));
A6: IC (Following (F,t)) in NIC ((F /. (IC (Comput (F,s,k)))),(IC (Comput (F,s,k)))) ;
A7: Comput (F,s,(k + 1)) = Following (F,t) by EXTPRO_1:4;
NIC ((F /. (IC (Comput (F,s,k)))),(IC (Comput (F,s,k)))) c= dom F by A1, A5, Def17;
hence S₁[k + 1] by A6, A7; :: thesis: verum

end;

A8: S₁[ 0 ] by A3, EXTPRO_1:3;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A8, A4); :: thesis: verum